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About 10,000 papers for “generating functions”

Generating Functions

Michael Drmota, Wojciech Szpankowski · 2023

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Generating functions

Sachkov Vi · Cambridge University Press eBooks · 1996

The generating functions considered in this chapter are important instruments for solving the so-called enumerative problems in combinatorial analysis. Enumerative problems arise if we need to be explicit about the number of ways of choosing particular elements from a finite set. The application of generating functions in this situation consists of establishing a correspondence between the elements of the set and the terms of the products of some series; the solution of enumerative problems is reduced, in fact, to finding a suitable method for the multiplication of these series. Under these conditions, the convergence of the series is not necessary, and it is natural to use a formal power series, assuming that the operations on them are properly defined. The formal power series, generally speaking, of several variables, are called the generating functions.

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Generating Functions

Robin Pemantle, Mark C. Wilson · Cambridge University Press eBooks · 2013

This chapter gives a crash course on generating functions and enumeration. For a more lengthy introduction, we recommend Wilf (2006). Chapter 14 of van Lint and Wilson (2001) also provides a fairly concise but readable treatment. Proofs for facts about formal power series may be found in Stanley (1997, Section 1.1). A comprehensive treatment of the relation between power series operations and corresponding combinatorial constructions on finite sets is the encyclopedic reference by Goulden and Jackson (2004). Chapters 1–3 of Flajolet and Sedgewick (2009) contain a very nice treatment as well. Bender and Williamson (1991) and Kauers and Paule (2011).

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Generating Functions

2001

This chapter contains sections titled: Generating Function Partial Fraction Decomposition Rule for p(x)/q(x) Where deg p(x) < deg q(x) Exercises 18

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Generating Functions

S. Ramanan · 2011

Power series The use of generating functions is one way of book-keeping. Let a : Z+ → C be any function, in other words, a sequence an of complex numbers. Then one associates to it the power series∑ anxn. The power series may not converge anywhere except x = 0, that is to say, its radius of convergence may be 0. If b is another such sequence, then the product of the two power series is associated to the function n → ∑i=ni=0 aibn−i. This is the Cauchy product of two sequences and so already we notice that associating the power series to the sequence has some advantage.

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Generating Functions

Andreas Klappenecker, Hyunyoung Lee · Undergraduate texts in mathematics · 2024

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Generating Functions

Henk Tijms · Cambridge University Press eBooks · 2018

Probability has applications in many areas of modern science, not to mention in our daily life. Its importance as a mathematical discipline cannot be overrated, and it is a fascinating and surprising topic in its own right. This engaging textbook with its easy-to-follow writing style provides a comprehensive yet concise introduction to the subject. It covers all of the standard material for undergraduate and first-year-graduate-level courses as well as many topics that are usually not found in standard texts, such as Bayesian inference, Markov chain Monte Carlo simulation, and Chernoff bounds.

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Generating functions

Cambridge University Press eBooks · 2007

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

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Generating Functions

Shahriar Shahriari · Cambridge University Press eBooks · 2021

Active student engagement is key to this classroom-tested combinatorics text, boasting 1200+ carefully designed problems, ten mini-projects, section warm-up problems, and chapter opening problems. The author – an award-winning teacher – writes in a conversational style, keeping the reader in mind on every page. Students will stay motivated through glimpses into current research trends and open problems as well as the history and global origins of the subject. All essential topics are covered, including Ramsey theory, enumerative combinatorics including Stirling numbers, partitions of integers, the inclusion-exclusion principle, generating functions, introductory graph theory, and partially ordered sets. Some significant results are presented as sets of guided problems, leading readers to discover them on their own. More than 140 problems have complete solutions and over 250 have hints in the back, making this book ideal for self-study. Ideal for a one semester upper undergraduate course, prerequisites include the calculus sequence and familiarity with proofs.

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Generating functions

William A. Schwalm · Morgan & Claypool Publishers eBooks · 2019

The generating function concept is introduced, and its usefulness is demonstrated by simple examples involving Hermite and Bessel functions. As a physical application, a one-dimensional chain of springs and weights is considered. A generating function is constructed for mechanical motion of the <i>n</i>th mass in the chain as a function of time, given that the zeroth mass is displaced and released from rest at time 0. The solutions generated thereby are the elementary solutions, or essentially Green functions in the time domain.

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*Generating functions

Graham J. G. Upton · 2025

Abstract This chapter is concerned with two types of generating functions: the probability generating function (PGF) and the moment generating function (MGF). Generating functions provide a compact way of presenting information about the distribution of a discrete random variable (using the PGF). The MGF is primarily used for continuous random variables. Generating functions can provide a simple way of solving problems that would otherwise be difficult. As an example, the MGF is used to prove the central limit theorem.

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